134 research outputs found
Worst and best irredundant sum-of-products expressions
In an irredundant sum-of-products expression (ISOP), each product is a prime implicant (Pl) and no product can be deleted without changing the function. Among the ISOPs for some function f, a worst ISOP (WSOP) is an ISOP with the largest number of Pls and a minimum ISOP (MSOP) is one with the smallest number. We show a class of functions for which the Minato-Morreale ISOP algorithm produces WSOPs. Since the ratio of the size of the WSOP to the size of the MSOP is arbitrarily large when it, the number of variables, is unbounded, the Minato-Morreale algorithm can produce results that are very far from minimum. We present a class of multiple-output functions whose WSOP size is also much larger than its MSOP size. For a set of benchmark functions, we show the distribution of ISOPs to the number of Pls. Among this set are functions where the MSOPs have almost as many Pls as do the WSOPs. These functions are known to be easy to minimize. Also, there are benchmark functions where the fraction of ISOPs that are MSOPs is small and MSOPs have many fewer Pls than the WSOPs. Such functions are known to be hard to minimize. For one class of functions, we show that the fraction of ISOPs that are MSOPs approaches 0 as n approaches infinity, suggesting that such functions are hard to minimiz
Comparison of the Worst and Best Sum-of-Products Expressions for Multiple-Valued Functions
Because most practical logic design algorithms produce irredundant sum-of-products (ISOP) expressions, the understanding of ISOPs is crucial. We show a class of functions for which Morreale-Minato's ISOP generation algorithm produces worst ISOPs (WSOP), ISOPs with the most product terms. We show this class has the property that the ratio of the number of products in the WSOP to the number in the minimum ISOP (MSOP) is arbitrarily large when the number of variables is unbounded. The ramifications of this are significant; care must be exercised in designing algorithms that produce ISOPs. We also show that 2/sup n-1/ is a firm upper bound on the number of product terms in any ISOP for switching functions on n variables, answering a question that has been open for 30 years. We show experimental data and extend our results to functions of multiple-valued variables
Area-power-delay trade-off in logic synthesis
This thesis introduces new concepts to perform area-power-delay trade-offs in a logic synthesis system. To achieve this, a new delay model is presented, which gives accurate delay estimations for arbitrary sets of Boolean expressions. This allows use of this delay model already during the very first steps of logic synthesis. Furthermore, new algorithms are presented for a number of different optimization tasks within logic synthesis. There are new algorithms to create prime irredundant Boo lean expressions, to perform technology mapping for use with standard cell generators, and to perform gate sizing. To prove the validity of the presented ideas, benchmark results are given throughout the thesis
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FACTPLA: Functional analysis and the complexity of testing programmable logic array
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.A computer aided method for analyzing the testability of Programmable Logic Arrays (PLAs) is described. The method, which is based on a functional verification approach, estimates the complexity of testing a PLA
according to the amount of single undetectable faults in the array structure.
An analytic program (FACTPLA) is developed to predict the above complexity without analyzing the topology of the array as such. Thus, the method is technology invariant
and depends only on the functionality of the PLA. The program quantitatively evaluates the effects of undetectable faults and produces some testability measures to manifest these effects. A testability profile for different PLA examples is provided and a number of suggestions for further research to establish definitely the usefulness of some functional properties for testing were made
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Deterministic, Efficient Variation of Circuit Components to Improve Resistance to Reverse Engineering
This research proposes two alternative methods for generating semantically equivalent circuit variants which leave the circuit\u27s internal structure pseudo-randomly determined. Component fusion deterministically selects subcircuits using a component identification algorithm and replaces them using a deterministic algorithm that generates canonical logic forms. Component encryption seeks to alter the semantics of individual circuit components using an encoding function, but preserves the overall circuit semantics by decoding signal values later in the circuit. Experiments were conducted to examine the performance of component fusion and component encryption against representative trials of subcircuit selection-and-replacement and Boundary Blurring, two previously defined methods for circuit obfuscation. Overall, results support the conclusion that both component fusion and component encryption generate more secure variants than previous methods and that these variants are more efficient in terms of required circuit delay and the power and area required for their implementation
Advanced Algorithms for VLSI: Statistical Circuit Optimization and Cyclic Circuit Analysis
This work focuses on two emerging fields in VLSI. The first is use of statistical formulations to tackle one of the classical problems in VLSI design and analysis domains, namely gate sizing. The second is on analysis of nontraditional digital systems in the form of cyclic combinational circuits.
In the first part, a new approach for enhancing the process-variation tolerance of digital circuits is described. We extend recent advances in statistical timing analysis into an optimization framework. Our objective is to reduce the performance variance of a technology-mapped circuit where delays across elements are represented by random variables which capture the manufacturing variations. We introduce the notion of statistical critical paths, which account for both means and variances of performance variation. An optimization engine is used to size gates with a goal of reducing the timing variance along the statistical critical paths. Circuit optimization is carried out using a gain-based gate sizing algorithm that terminates when constraints are satisfied or no further improvements can be made. We show optimization results that demonstrate an average of 72% reduction in performance variation at the expense of average 20% increase in design area.
In the second part, we tackle the problem of analyzing cyclic circuits. Compiling high-level hardware languages can produce circuits containing combinational cycles that can never be sensitized. Such circuits do have well-defined functional behavior, but wreak havoc with most tools, which assume acyclic combinational logic. As such, some sort of cycle-removal step is usually necessary. We present an algorithm able to quickly and exactly characterize all combinational behavior of a cyclic circuit. It used a combination of explicit and implicit methods to compute input patterns that make the circuit behave combinationally. This can be used to restructure the circuit into an acyclic equivalent, report errors, or as an optimization aid. Experiments show our algorithm runs several orders of magnitude faster than existing ones on real-life cyclic circuits, making it useful in practice
Synthesis of Quantum Logic Circuits
We discuss efficient quantum logic circuits which perform two tasks: (i)
implementing generic quantum computations and (ii) initializing quantum
registers. In contrast to conventional computing, the latter task is nontrivial
because the state-space of an n-qubit register is not finite and contains
exponential superpositions of classical bit strings. Our proposed circuits are
asymptotically optimal for respective tasks and improve published results by at
least a factor of two.
The circuits for generic quantum computation constructed by our algorithms
are the most efficient known today in terms of the number of expensive gates
(quantum controlled-NOTs). They are based on an analogue of the Shannon
decomposition of Boolean functions and a new circuit block, quantum
multiplexor, that generalizes several known constructions. A theoretical lower
bound implies that our circuits cannot be improved by more than a factor of
two. We additionally show how to accommodate the severe architectural
limitation of using only nearest-neighbor gates that is representative of
current implementation technologies. This increases the number of gates by
almost an order of magnitude, but preserves the asymptotic optimality of gate
counts.Comment: 18 pages; v5 fixes minor bugs; v4 is a complete rewrite of v3, with
6x more content, a theory of quantum multiplexors and Quantum Shannon
Decomposition. A key result on generic circuit synthesis has been improved to
~23/48*4^n CNOTs for n qubit
Optimal Approximate Minimization of One-Letter Weighted Finite Automata
In this paper, we study the approximate minimization problem of weighted
finite automata (WFAs): to compute the best possible approximation of a WFA
given a bound on the number of states. By reformulating the problem in terms of
Hankel matrices, we leverage classical results on the approximation of Hankel
operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory.
We solve the optimal spectral-norm approximate minimization problem for
irredundant WFAs with real weights, defined over a one-letter alphabet. We
present a theoretical analysis based on AAK theory, and bounds on the quality
of the approximation in the spectral norm and norm. Moreover, we
provide a closed-form solution, and an algorithm, to compute the optimal
approximation of a given size in polynomial time.Comment: 32 pages. arXiv admin note: substantial text overlap with
arXiv:2102.0686
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